Centroids of Type-1 and Type-2 Fuzzy Sets When Membership Functions Have Spikes

Centroids are practically important in type-1 and type-2 fuzzy logic systems as a method of defuzzification and type reduction. However, computational problems arise when membership functions (MF) have singleton spikes. The novel thresholding aggregation operators that were described in our companion paper “ New Classes of Threshold Aggregation Functions Based Upon the Tsallis q-Exponential with Applications to Perceptual Computing” produce such MFs with spikes. Such spikes may occur when modeling concepts defined on a real-valued domain, and they are also formed in unions of fuzzy sets in which some have MFs with discrete support and others have support defined on an interval. This paper presents a modified definition of the centroid of a fuzzy set that avoids the computational problems associated with the usual definition and reduces to this definition when MFs are continuous and normal (i.e., of unit height) on some interval. We also present an enhanced Karnik-Mendel-type algorithm to compute the modified centroid of interval type-2 fuzzy sets whose MFs have spikes.